Classically, a single weight on an interval of the real line leads to
moments, orthogonal polynomials and tridiagonal matrices. Appropriately
deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda
lattice and tau-functions, expressed as Hermitian matrix integrals.
This paper is concerned with a sequence of t-perturbed weights, rather than
one single weight. This sequence leads to moments, polynomials and a (fuller)
matrix evolving according to the discrete KP-hierarchy. The associated
tau-functions have integral, as well as vertex operator representations.
Among the examples considered, we mention: nested Calogero-Moser systems,
concatenated solitons and m-periodic sequences of weights. The latter lead to
2m+1-band matrices and generalized orthogonal polynomials, also arising in the
context of a Riemann-Hilbert problem.
We show the Riemann-Hilbert factorization is tantamount to the factorization
of the moment matrix into the product of a lower- times upper-triangular
matrix.